We study the response of pinned flux lattices, under small perturbations in the driving force, below^ and close to the pinning-depinning transition. For driving Lorentz forces below Fc (the depinning force at which the whole flux lattice slides), the system has instabilities against small force increases, with a power-law distribution characteristic of self-organized criticality. Specifically, Z>(t/)-^ ~'^, where d is the displacement of a flux line after a very small force increase. We also study the initial stages of the motion of the lattice once the driving force overcomes the pinning forces.

The dynamics of an individual magnetic moment is studied through the Landau-Lifshitz equation with a periodic driving in the direction perpendicular to the applied field. For fields lower than the anisotropy field and small values of the perturbation amplitude we have observed the magnetic moment bistability. At intermediate values we have found quasiperiodic bands alternating with periodic motion. At even larger values a chaotic regime is found. When the applied field is larger than the anisotropy one, the behavior is periodic with quasiperiodic regions. Those appear periodically in the amplitude of the oscillating field. Also, even for low values of the driving force, the moment is not parallel to the applied field. Traditionally, the study of the dynamics governed by the Landau-Lifshitz equation is related to the ferromagnetic resonance problems 1 . Recently, the spin dynamics has become also important in other physical phenomena relevant to technological applications, such as, e.g., magnetic recording processes 2,3 due to a continuous increase of the magnetic recording density together with the writing frequency 3,4 . The dynamical micromagnetic calculations have provided a useful tool in studying such important media characteristic as dynamical coercitivity 5 . In spite of the fact that the Landau-Lifshitz equation is widely used in micromagnetic calculations, up to our knowledge, no systematic study of its dynamics exists in the literature. Let us recall here that this equation is nonlinear, and in some regime one may expect a highly complicated dynamics similar to one arising for an externally driven pendulum. As an example, we can mention the nonlinear stochastic resonance behavior of an individual magnetic moment 6 . The purpose of this paper is to present a systematic study of nonlinear dynamics governed by the Landau-Lifshitz equation, including its bifurcation diagram and stability properties. This would be very useful when analyzing results obtained from large simulations of coupled Landau-Lifshitz (LL) equations. There, and for some values of the coupling parameters, the individual characteristics of each magnetic moment may play an important role in the collective behavior. Eventually, when enough of these moments are coupled, a description in terms of magnons would be possible 11Also the subject of chaos in magnetic materials 7 is not new. It has been studied in YIG, both experimental and theoretically, through spin wave descriptions 8,9 . In different driving regime, the chaotic behavior can arise in magnetostrictive wires and ribbons due to the magnetoelastic coupling 10 . In the original form (Landau-Lifshitz-Gilbert 12 ) the equation may be written as:or in the more practical form (Landau-LifshitzWhere g is the local giromagnetic factor, η is the damping, M 0 the saturation magnetization, M is the tridimensional local continuous magnetization, whose module is conserved (M · M = M 0 ), and H eff the effective field:Here H ext is the external magnetic field, β is the anisotropy coefficient, n i...

Abstract. The El Farol bar model, proposed to study the dynamics of competition of agents in a variety of contexts (W. B. Arthur, Amer. Econ. Assoc. Pap. and Proc. 84, 406 (1994)) is studied. We characterize in detail the three regions of the phase diagram (efficient, better than random and inefficient) of the simplest version of the model (D. Challet and Y.-C. Zhang, Physica A, 246, 407 (1997)). The efficient region is shown to have a rich structure, which is investigated in some detail. Changes in the payoff function enhance further the tendency of the model towards a wasteful distribution of resources.

¿From a very simple description of vortices and pinning centers and without using any electrodynamical assumptions, we obtain nonlinear density profiles of vortices in type-II dirty superconductors that result from raising or lowering an external magnetic field. The results confirm a modified Bean model description of these systems, and in particular, follow the Kim empirical form that relates the current inside the material to the local magnetic field. We also obtain realistic magnetization hysteresis loops and examine the discrete evolution of the density profiles in our systems. This evolution is not continuous, but takes place by the occurrence of avalanches of vortices promoted by the addition or extraction of vortices from the edges of the system. 74.60.Ge, 05.60.+w 1